In total, we consider 20 distinct families of logical theories (each family containing a different theory for each possible underlying alphabet \(\Sigma \) ), which are introduced in detail in Section . Taking inspiration from [1]} (and partly from [2]}, where relation-definability by logics over strings was studied in a database-theory centered framework),
we study these logics from a formal languages perspective by looking at the set of values a variable may take while preserving satisfiability of a formula. Specifically, given a formula \(f\)  from a quantifier-free logical theory \(\mathfrak {T}\) , we say that the language expressed by a variable \(x\)  occurring in \(f\)  is the set of concrete values \(w\)  such that substituting \(x\)  for \(w\)  in \(f\)  yields a satisfiable formula. In the general case, we can think of the property that the formula \(f\)  defines via the variable \(x\) . However, since we consider logics in which \(x\)  is substituted for finite strings, we get a formal language.
